A text book's proof

For discussing Olympiad Level Algebra (and Inequality) problems
Hasib
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A text book's proof

Unread post by Hasib » Tue Dec 14, 2010 4:25 pm

I think it's very easy to prove the following rule $A\cup (B\cap C)$ easier way than our text book!
The proof is here:
\[A\cup (B\cap C)\]
\[= \{x:x\in A\text{ or }x\in (B\cap C)\}\]
\[= \{x:x\in A\text{ or }(x\in B\text{ and }x\in C)\}\]
\[= \{x:(x\in A\text{ or }x\in B)\text{ and }(x\in A or x\in C)\}\]
\[= \{x:x\in A\cup B\text{ and }x\in A\cup C\}\]
\[= \{x:x\in (A\cup B)\cap(A\cup C)\}\]
\[=(A\cup B)\cap (A\cup C)\]
\[[proved]\]
so, why our text book contains so hard proof?
Last edited by Hasib on Wed Dec 15, 2010 6:23 pm, edited 7 times in total.
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Hasib
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Re: A text book's proof

Unread post by Hasib » Tue Dec 14, 2010 4:30 pm

Please, Any one can say me why the space doesn't working in upper equetions??
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Moon
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Re: A text book's proof

Unread post by Moon » Tue Dec 14, 2010 6:28 pm

you have to indicate spacing if you want to put spaces inside equation.
Form http://www.artofproblemsolving.com/Wiki ... TeX:Layout
However, if you do need to tweak the spacing in math mode, there are some special commands:

\, a small space
\: a medium space
\; a large space
\quad a really large space
\qquad a huge space
\! a negative space (moves things back to the left)
Here are examples of these in action:
BTW if you want to write text inside equation you better use \text{your text}
For example x \in A\text{ or } x\in B gives $x \in A\text{ or } x\in B$
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin

Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.

Hasib
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Re: A text book's proof

Unread post by Hasib » Tue Dec 14, 2010 9:26 pm

Moon wrote:you have to indicate spacing if you want to put spaces inside equation.
Form http://www.artofproblemsolving.com/Wiki ... TeX:Layout
However, if you do need to tweak the spacing in math mode, there are some special commands:

\, a small space
\: a medium space
\; a large space
\quad a really large space
\qquad a huge space
\! a negative space (moves things back to the left)
Here are examples of these in action:
BTW if you want to write text inside equation you better use \text{your text}
For example x \in A\text{ or } x\in B gives $x \in A\text{ or } x\in B$

thanks vaia a lot!!!
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Re: A text book's proof

Unread post by Zzzz » Wed Dec 15, 2010 8:37 am

hasib.mo wrote:I think it's very easy to prove the following rule $A\cup (B\cap C)$ easier way than our text book!
The proof is here:
...
বইয়ের প্রমাণ আর তোমার প্রমানে তো তেমন পার্থক্য পাইলাম না। বইয়ে জিনিসটা বিপরীত দিক থেকেও করা আছে, যেটা করা উচিৎ।

btw, এখানে কিছু latex code দিলাম যেটা মনে হয় তোমার কাজে লাগবেঃ
\Rightarrow = $\Rightarrow$
\Leftrightarrow = $\Leftrightarrow$
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Re: A text book's proof

Unread post by Zzzz » Wed Dec 15, 2010 8:42 am

ও আচ্ছা, বইয়ে এইটা সরাসরি প্রমাণ নাই। $A\cup(B\cup C) = (A\cup B)\cup (A\cup C)$ এইভাবে প্রমান করে লিখছে তোমার করা প্রমানটা একইভাবে করা যায়।
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Hasib
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Re: A text book's proof

Unread post by Hasib » Wed Dec 15, 2010 9:02 am

Ha. The text book contains the proof with two side, and they use the method of subset($\subset$) bt, i use that logic bt, in a changed form(i use directly equal sign!). So, i think i don't need to do the proof in two side with the changed form. :) thanks!
Bt, { } (bracket) isn't working in the dollar sign! What will i do?! And thanks again!
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Re: A text book's proof

Unread post by Moon » Wed Dec 15, 2010 9:18 am

bracket is a part of LaTeX command...so you have to use \{x\} for $\{x\}$
A few words for bracket: use \left ( \sum x^2 \right)^2 for bigget size bracket..I mean \left ( \right) \[ \left ( \sum x^2 \right)^2\]
ex:
\[\left\{ \int x^2\, dx \right \}^2=1\]
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Please install LaTeX fonts in your PC for better looking equations,
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Re: A text book's proof

Unread post by Zzzz » Wed Dec 15, 2010 9:28 am

You can not use $=$ sign there. "$x:x \in ...$' it means $x$ such that $x$ belongs to... . If you write \[A\cup (B \cap C) = x:x\in ... \] it means that set is equal to $x$. You know, that set is not equal to that $x$.
$\Rightarrow$ means 'implies that'
\[A\cup (B \cap C) \Rightarrow x:x\in A\cup (B \cap C)\] এটা ঠিক আছে। $A\cup (B \cap C) $ বলতে এমন সব $x$ কে বুঝায় যেন $x\in A\cup (B \cap C)$
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Hasib
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Re: A text book's proof

Unread post by Hasib » Wed Dec 15, 2010 6:14 pm

Moon wrote:bracket is a part of LaTeX command...so you have to use \{x\} for $\{x\}$
A few words for bracket: use \left ( \sum x^2 \right)^2 for bigget size bracket..I mean \left ( \right) \[ \left ( \sum x^2 \right)^2\]
ex:
\[\left\{ \int x^2\, dx \right \}^2=1\]

oh, thankS, but, isn't the first bracket a La TeX command??
Last edited by Hasib on Wed Dec 15, 2010 6:27 pm, edited 1 time in total.
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